Cosheaves and connectedness in formal topology

peer-00821313, version 1 - 9 May 2013
a ′ ≤ γ a. Since a ′ ⊳ s − (a, γ) ↓ s − V ⊳ s − ((a, γ) ↓ V ), and a ′ is positive,
(a, γ) ↓ V is inhabited.
Now suppose that (b i , δ i ) ∈ (a, γ) ↓ V (i = 1, 2); we must show that (b 1 , δ 1 ) ∼
(b 2 , δ 2 ). We can find (b ′ i , δ′ i ) ≤ (b i, δ i ) with b ′ i ≤δi b i so that b ′ i ≤γ a; also,
(b i , δ i ) ∼ (b ′ i , δ′ i ). Hence without loss of generality we can assume b i ∈ s − ((a, γ) ↓
V ). Since a ⊳ ⋃ {≤ α a | α ∈ π 0 (a)}, there is a connection from (b 1 , δ 1 ) to
(b 2 , δ 2 ) in ⋃ {≤ α a | α ∈ π 0 (a)}. If we have (c 1 , ε 1 ) ≤ (c 2 , ε 2 ) with c i ≤ αi a then
α 1 = (π 0 ) c1,a(ε 1 ) = (π 0 ) c2,a(ε 2 ) = α 2 , and it follows that the connection from
(b 1 , δ 1 ) to (b 2 , δ 2 ) must be in ≤ γ a. If (c, ε) is an element of the connection, then
because (≤ γ a) ⊳ s − V we can find (c ′ , ε ′ ) ≤ (c, ε) with c ′ ∈ s − V . Suppose now
we have (c 1 , ε 1 ) ≤ (c 2 , ε 2 ) as a link in the connection from (b 1 , δ 1 ) to (b 2 , δ 2 ).
Then there is a connection from (c ′ 1, ε ′ 1) to (c ′ 2, ε ′ 2) in c 2 ↓ s − V ⊆ s − ((a, γ) ↓ V )
and so (c ′ 1, ε ′ 1) ∼ (c ′ 2, ε ′ 2). Putting these together, and remembering that (b 1 , δ 1 )
and (b 2 , δ 2 ) are already in (a, γ) ↓ V , we find (b 1 , δ 1 ) ∼ (b 2 , δ 2 ).
This completes the proof of part (1). For part (4), since π 0 ′ is a terminal
cosheaf for Y , and p is a homeomorphism, Cosh p(π 0) ′ is a terminal cosheaf for
X.
In Proposition 31 we shall see the above construction of ( ∑ a∈P π 0(a), ≤, ⊳)
and p as a special example of the “complete spread” construction (for cosheaves
other than π 0 ), which can be described more generally in the inductively generated
case.
Theorem 24. A formal topology is locally connected iff it is homeomorphic to
a formally locally connected formal topology.
Proof. The ⇒ direction is Proposition 23. The ⇐ direction comes from Corollary
22.
We shall next show how to find connected components for arbitrary opens
in a locally connected formal topology.
Definition 25. Let X = (P, ≤, ⊳) be locally connected, and suppose A ⊆ P
and γ ∈ π 0 (A) = colim a∈↓A π 0 (a). Define
≤ γ A = ⋃ {≤ δ a | γ = in a (δ)}.
Proposition 26. Let X = (P, ≤, ⊳) be locally connected, and suppose A ⊆ P .
1. A ⊳ ⋃ γ∈π 0(A) (≤γ A).
2. The opens ≤ γ A (γ ∈ π 0 (A)) are pairwise disjoint.
3. Each ≤ γ A is connected.
Proof. (1) This follows from the fact that a ⊳ ⋃ γ∈π 0(a) ≤γ a.
(2) Suppose b ∈ (≤ γ1 A) ↓ (≤ γ2 A). Then b ≤ δi a i with in ai (δ i ) = γ i . If
ε ∈ π 0 (b) then (b, ε) ≤ (a i , δ i ) (i = 1, 2) so γ i = in b (ε) and γ 1 = γ 2 . Hence
(≤ γ1 A) ↓ (≤ γ2 A) ⊳ {d ∈ P | γ 1 = γ 2 }.
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